Mass tells spacetime how to curve — the rubber sheet made rigorous
Einstein's revolutionary insight: gravity is not a force — it is the curvature of spacetime. Mass and energy tell spacetime how to curve, and curved spacetime tells objects how to move. A falling apple follows the straightest possible path (a geodesic) through curved spacetime.
This page builds the geometric machinery: the metric tensor that measures distances, the Riemann tensor that quantifies curvature, and the visual intuition of spacetime bending under mass.
Place masses on a 3D grid and watch spacetime deform. The grid represents two spatial dimensions — the depression around each mass shows how spacetime curves. Glowing particles follow geodesics (the "straightest" paths in curved space), showing how gravity emerges from geometry.
Try it: Click to place masses, drag to move them. Launch particles and watch them orbit, scatter, or plunge — all without any "force" acting on them. They're simply following the straightest path through curved spacetime.
The metric tensor g_μν is the fundamental object of general relativity. It tells you how to measure distances in curved spacetime. In flat spacetime, it's the Minkowski metric. Near a massive object, the metric components change — clocks run slower and rulers stretch.
g_ij = diag(1, 1). Uniform grid, equal cell areas.
Curvature has a precise mathematical meaning: when you parallel-transport a vector around a closed loop on a curved surface, it comes back rotated. The amount of rotation tells you the curvature enclosed by the loop.
Larger loops enclose more area, causing a greater rotation angle. On a sphere of curvature K, the holonomy equals the enclosed solid angle.
Two nearby geodesics that start parallel will converge or diverge in curved spacetime. This geodesic deviation is governed by the Riemann curvature tensor — the mathematical object that fully encodes the curvature of spacetime.
The classic "well in a rubber sheet" — but done properly. This is Flamm's paraboloid: a mathematically exact embedding of the Schwarzschild spatial geometry into 3D Euclidean space. The depth represents how much spatial distances are stretched near the mass.